THE SPINNING MEMBRANE, SUPER-SU(∞)-GAUGE THEORIES AND MOYAL BRACKETS

Abstract

Several important topics concerning the membrane and its symmetries are discussed. The fact that a space–time-independent Lagrangian density for a gauge-field configuration of a (d – 1)-dimensional SU (∞) super Yang–Mills theory, reduced to one dimension (time), is equivalent to a Green–Schwarz formalism of the Euclidean Eguchi–Schild string action in d – 1 dimensions, naturally raises the question whether one can construct a Neveu–Ramond–Schwarz analog. The answer is in the negative; the world-sheet supersymmetric extension of the Eguchi–Schild action for the string cannot be viewed as a classical-vacuum configuration of a super-SU (∞)- gauge theory. For the second topic we construct a "supersymmetry" charge operator, Qf, which plays the role of a residual fermionic symmetry, for fixed time, of the light-cone spinning membrane. It is explicitly shown how the Yang–Mills type of actions and, in particular, the ones for vacuum-field configurations, associated with Q(∞) supergauge theories, are invariant under both Qf "supersymmetry" and the superalgebra of area-preserving superdiffeomorphisms of the light-cone spinning torus membrane, Q(∞). More general actions can be constructed which are invariant under deformations of this superalgebra. In this case the ordinary (graded) Poisson brackets are replaced by super Moyal brackets. Finally, we conjecture why these actions, in analogy with what happens with the light-cone supermembrane, should correspond to a superfiber bundle (over space–time) formulation of the supersymmetric-gauge quantum-mechanical models (SGQMM's) of Flume and Baake et al.; with the general supergroup of trigonometric structure constants of Fairlie, Fletcher and Zachos as the structure supergroup of the superfiber. To support our concluding conjecture, preliminary steps are outlined which are necessary in order to fix the light-cone gauge for the spinning-membrane action. We discuss why the Qf "supersymmetry" (the remnant world-volume light-cone local supersymmetry) and the Q(∞) supergauge transformations must arise as its residual symmetries.